The resonant spectrum of the acoustic crystal, as computed with the finite-element based software COMSOL [1]}, is reported in Fig. REF (e) as function of \(\theta \) . Since the simulations were for a finite crystal, some of the bulk gaps are contaminated by boundary spectrum. Additional model calculations with periodic boundary conditions and for larger crystals are reported in Supplementary Information. At \(\theta =0\) , the crystal is periodic and, as expected, the spectrum contains bands that evolve from the discrete modes of the individual resonators. These bands don't share any dynamical features, hence the spectral gaps separating them are all trivial [see the label in Fig. REF (e)]. As the parameter \(\theta \)  is turned on, the bands of spectrum become fragmented and a large number of spectral gaps develop. Qualitatively, the spectra resembles the Hofstadter butterfly [2]} and, as we shall see below, the spectral gaps carry 2nd and 1st Chern numbers.
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